The rings where zero-divisor polynomials have zero-divisor coefficients

Abstract

Based on McCoy’s theorem on commutative rings, Nielsen called a ring $R$ if the equation $f\left(x\right)g\left(x\right)=0$ implies $f\left(x\right)c=0$ for some nonzero element $c$ in $R$, where $f\left(x\right)$ and $g\left(x\right)$ are nonzero polynomials in $R\left[x\right]$. In this paper, a class of rings is introduced and called ZPZC rings, containing McCoy rings, and then their properties are investigated. Also, associations are found between ZPZC rings and other related rings. Moreover, several extensions of ZPZC rings are studied, including matrix rings, trivial extensions, Hochschild extensions and classical quotient rings.